The embodiments described herein relate to an image reconstruction method and an X-ray CT (Computed Tomography) apparatus.
When a CT image is reconstructed based on projection data acquired by an X-ray CT scan, pixel data of each pixel on a reconstruction plane, corresponding to the CT image has heretofore been reconstructed using general formulas for backprojection arithmetic operation including weighting, such as expressed in the following equations (refer to, for example, “A three-dimensional weighted cone beam filtered backprojection (CB-FBP) algorithm for image reconstruction in volumetric CT under a circular source trajectory”, Physics In Medicine And Biology, 50 (2005), P.3889-3905, Xiangyang Tang, Jiang Hsieh, Akira Hagiwara, et al.).
                              f          ⁡                      (                          x              ,              y              ,              z                        )                          =                              π                          (                                                β                  max                                -                                  β                  min                                            )                                ⁢                                    ∫                              β                min                                            β                max                                      ⁢                                          Fi                                                                                                    (                        Fi                        )                                            2                                        +                                          Z                      2                                                                                  ⁢                              ω                ⁡                                  (                                      β                    ,                    γ                                    )                                            ⁢                                                s                  ~                                ⁢                                                                  (                                  α                  ,                  β                  ,                  γ                                )                            ⁢                              ⅆ                β                                                                        (        1        )                                                          ⁢                                            s              ~                        ⁢                                                  (                          α              ,              β              ,              γ                        )                    =                                    s              ⁢                                                          (                              α                ,                β                ,                γ                            )                        ⊗                          g              ⁡                              (                γ                )                                                                        (        2        )            
As shown above, f(x, y, z) in equation (1)indicates pixel data of a pixel P(x, y, z) on a reconstruction plane RP, s(α, β, γ) indicates projection data based on an X-ray beam that passes through the pixel P(x, y, z) on the reconstruction plane RP, g(γ) indicates a reconstruction function, a symbol {circle around (X)} with X being surrounded with ∘ indicates a convolution operator, α, indicates a cone angle formed as viewed from a plane that passes through an X-ray focal point F and a center line in a slice direction, of an X-ray detector D, β indicates a view angle position on a circular orbit, of the X-ray focal point F, γ indicates an angle in a channel direction as viewed from a central axis of a fan-beam X ray, Fi indicates the distance from the X-ray focal point F to a rotational center axis Ic of the X-ray focal point F, Z indicates a predetermined value that depends on a z coordinate of the pixel P, and ω (β, γ) indicates a weighting coefficient (weighting factor) by which the projection data s (α, β, γ) is multiplied. Incidentally, (βmax−βmin=π+2γm (where γm=half of a fan angle of a fan-shaped X-ray beam) in the case of half reconstruction, and βmax−βmin=2π or 2π+2γm in the case of full reconstruction.
Incidentally, weighting coefficients by which plural projection data substantially identical or opposite to one another in view angle direction θ, of the projection data s (α,β, γ) are multiplied are set so that the sum thereof reaches 1.
Now consider where the X-ray CT scan is taken as a helical scan. In this case, there can be considered as projection data s (α, β, γ), first projection data s (αdi, (βdi, γdi) at which an angle direction at the time that a corresponding X-ray beam passes through a pixel P from one plane side of a reconstruction plane RP and is projected onto the reconstruction plane RP, is a view angle direction θi, and second projection data s (αci, βci, γci) at which an angle direction at the time that a corresponding X-ray beam passes through the pixel P from the other plane side of the reconstruction plane RP and is projected onto the reconstruction plane RP is a view angle direction opposite to the view angle direction θi. At this time, a weighting coefficient ω (βdi, γdi) by which the first projection data s (αdi, βdi, γdi) is to be multiplied, and a weighting coefficient ω (βci, γci) by which the second projection data s (αci, βci, γci) is to be multiplied, can be determined in accordance with, for example, the following equations:
                                          ω            ⁢                          (                                                                    β                    d                                    ⁢                  i                                ,                                                      γ                    d                                    ⁢                  i                                            )                                =                                    Δ              ⁢                                                          ⁢                              Z                c                            ⁢              i                                                      Δ                ⁢                                                                  ⁢                                  Z                  c                                ⁢                i                            +                              Δ                ⁢                                                                  ⁢                                  Z                  d                                ⁢                i                                                    ,                              ω            ⁡                          (                                                                    β                    c                                    ⁢                  i                                ,                                                      γ                    c                                    ⁢                  i                                            )                                =                                    Δ              ⁢                                                          ⁢                              Z                d                            ⁢              i                                                      Δ                ⁢                                                                  ⁢                                  Z                  c                                ⁢                i                            +                              Δ                ⁢                                                                  ⁢                                  Z                  d                                ⁢                i                                                                        (        3        )                                                      ω            ⁡                          (                                                                    β                    d                                    ⁢                  i                                ,                                                      γ                    d                                    ⁢                  i                                            )                                +                      ω            ⁢                          (                                                                    β                    c                                    ⁢                  i                                ,                                                      γ                    c                                    ⁢                  i                                            )                                      =        1                            (        4        )            
In the equations, ΔZdi indicates the distance in a z direction between a reference position (for example, a position of a point where a straight line formed by connecting an X-ray focal point F and the center of an X-ray detector 24 and a rotational axis Ic of an X-ray generator and detector system 41 intersect) of the X-ray generator and detector system at the time that the first projection data s (αdi, βdi, γdi) is acquired, and a reconstruction plane RP, and ΔZci indicates the distance in the z direction between the reference position of the X-ray generator and detector system at the time that the second projection data s (αci, βci, γci) is acquired, and the reconstruction plane RP. Incidentally, the equation (4) is described in the sense of confirmation.